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Homesemisimple ring
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semisimple ring
A ring $R$ is (left) semisimple if it satisfies one of the following equivalent statements:
1. All left $R$modules are semisimple.
2. All finitelygenerated left $R$modules are semisimple.
3. All cyclic left $R$modules are semisimple.
4. The left regular $R$module ${}_{R}R$ is semisimple.
5. All short exact sequences of left $R$modules split.
The last equivalent condition offers another homological characterization of a semisimple ring:

A ring $R$ is (left) semisimple iff all of its left modules are projective.
A more ringtheorectic characterization of a (left) semisimple ring is:

A ring is left semisimple iff it is semiprimitive and left artinian.
In some literature, a (left) semisimple ring is defined to be a ring that is semiprimitive without necessarily being (left) artinian. Such a ring (semiprimitive) is called Jacobson semisimple, or Jsemisimple, to remind us of the fact that its Jacobson radical is (0).
Relating to von Neumann regular rings, one has:

A ring is left semisimple iff it is von Neumann regular and left noetherian.
The famous WedderburnArtin Theorem states that a (left) semisimple ring is isomorphic to a finite direct product of matrix rings over division rings.
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